Two-Dimensional Point-Group Demo

This page is a demo of the 2D discrete point groups or rosette groups.
A point group is a group of rotations and/or reflections about some point.
Some objects are symmetric with respect to some groups.
That means that they look alike when group members are applied to them.
This demo uses polygons, which are symmetric under these groups.
This demo works by applying members of the polygons' symmetry groups to those polygons;
you can see that they look the same afterward.

WARNING! This demo uses inline SVG, so it may not work in some web browsers.

Demo

The two-dimensional point groups can be subdivided in two ways:

Continuous vs. discrete

Pure rotation (cyclic) vs. rotation and reflection (dihedral)

which combine into these possibilities:

Pure Rotation

With Reflection

Continuous

SO(2)

O(2)

Discrete

C(n)

D(n)

Order (# Members)

n

2n

There are n rotations and n reflections.
SO is "special orthogonal", O is "orthogonal".

Cases equivalent to the cyclic group Z(n) for some n, the group of addition modulo n:

C(n) = Z(n)

D(1) = Z(2)

D(2) = Z(2)*Z(2)

D(3) is the smallest nonabelian (non-commutative) group,
and D(n) is nonabelian for all n at least 3.
However, the cyclic group is abelian.